The Glasgow Warriors second row was originally named among the replacements for Scotland's meeting with world champions South Africa, but was forced to withdraw just hours before kick-off in Pretoria. He was replaced on the bench by fellow Glasgow forward Max Williamson.
Following Scotland's 42-28 defeat, head coach Gregor Townsend confirmed that Samuel will need surgery.
Townsend said: "Alex Samuel will be getting surgery next week when we come back, so he'll be out a number of weeks.
"It's really unfortunate for him, he was due to be on the bench. But I have to say, Max Williamson stepped up today, came off the bench, made a big impact, and I felt Gregor Brown and Scott Cummings went really well as a second-row pairing."
Samuel made a substitute appearance in Scotland's victory over Argentina last weekend, and Townsend admitted the injury setback was disappointing. However, he believes the lock will "definitely" return to contention once he has recovered from his knee problem.
Scotland are expected to make several changes for Saturday's fixture at Murrayfield against Fiji, who suffered a 73-8 defeat to England in Liverpool.
Captain Rory Darge is also a doubt after failing a head injury assessment during the match in Pretoria.
Providing an update on the squad's condition after the game, Townsend was optimistic that there were no additional major injury concerns.
He added: "Often you find out after the game that there are injuries, but there was a lot of cramp with a few guys.
"Ben White had 80 minutes which is a huge effort, the way we play and what we demand of our scrum-half. And there was others, obviously Jack Dempsey playing 80.
"I don't think we have anything serious."